Finite Group Databasebeta

← #13412653

Subgroups — #13412653

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89893C211yesmaximal, characteristic
2C2×#89474C211yesmaximal, characteristic, complemented
3#11730939C211yesmaximal, characteristic
4#11730938C211yesmaximal, characteristic
5C7×#1753C211yesmaximal, characteristic
6#11729742C211yesmaximal, characteristic
7#11727683C211yesmaximal, characteristic
8#11727682C211yesmaximal, characteristic
9#11727573C211yesmaximal, characteristic, complemented
10#11727524C211yesmaximal, characteristic, complemented
11#11727518C211yesmaximal, characteristic, complemented
12#11726394C211yesmaximal, characteristic, complemented
13#11726393C211yesmaximal, characteristic, complemented
14#11726388C211yesmaximal, characteristic, complemented
15#11726387C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×#6441—21no—
20C2×#6433C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C4×C7×C22⋊C4C2211yescharacteristic
26#89984C2211yescharacteristic
27#89984C2211yescharacteristic
28#89893—24no—
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6391C2211yescharacteristic
34C2×#6391C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6391C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38C2×#6391C2211yescharacteristic
39#89722—21no—
40#89721—21no—
41#89720—21no—
42#89707C2211yescharacteristic
43#89707C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48#89474—24no—
49#89433—21no—
50#89428—21no—
51#89426—21no—
52#89419—21no—
53C7×#336C2211yescharacteristic
54C7×#336C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89346C2211yescharacteristic
61#26078—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C2×C7×M16Q814yescomplemented
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C2×C7×C22⋊C4D411yescharacteristic
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79#6489—22no—
80#6489—41no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—41no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×C4×Dic7—21no—
104#6441—44no—
105#6433—22no—
106#6433—41no—
107#6432—22no—
108#6432—41no—
109C7×C22⋊C8Q814yescomplemented
110C7×#97—22no—
111C4×C7×C8—22no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120#6392—22no—
121#6392—22no—
122#6392—22no—
123#6392D414yes—
124#6391—22no—
125#6391—22no—
126#6391—22no—
127#6391—22no—
128#6391—41no—
129#6391—41no—
130C2×#501—71no—
131C2×#428—71no—
132#2441—71no—
133#2440—71no—
134#2084—71no—
135#2083—71no—
136#2059—71no—
137#2058—71no—
138#1943—71no—
139#1942—71no—
140#1854—71no—
141#1853—71no—
142#1844—71no—
143#1809—71no—
144#1753D711yescharacteristic
145C24×C7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2×Q812yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—42no—
167C22×Dic7—42no—
168C22×Dic7—81no—
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14—81no—
172C7×M16—28no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C2×D412yes—
180C2×C7×C8C2×Q812yes—
181C2×C7×C8C2×Q812yes—
182C2×C7×C8C2×Q812yes—
183C2×C7×C8C2×Q812yes—
184C2×C7×C8#4112yes—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C22⋊Dic7—42no—
192C22⋊Dic7—42no—
193C22⋊Dic7—81no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—42no—
209C4⋊Dic7—42no—
210C4⋊Dic7—42no—
211C4⋊Dic7—42no—
212C4⋊Dic7—81no—
213C4⋊Dic7—81no—
214C4⋊Dic7C2×D412yes—
215C4⋊Dic7C2×D412yes—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
219Dic7⋊C4—42no—
220Dic7⋊C4—42no—
221Dic7⋊C4—81no—
222C4×Dic7—42no—
223C22×M16C2×D711yescharacteristic
224C2×C4⋊Q8—71no—
225C2×#125—71no—
226C2×C22⋊Q8—141no—
227C2×#117—71no—
228C22×C4⋊C4—71no—
229#501—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106—71no—
240C2×#106—71no—
241C2×C22⋊C8C2×D711yescharacteristic
242C2×#97C2×D711yescharacteristic
243C2×C4×C8C2×D711yescharacteristic
244#399—141no—
245#398—141no—
246#396—141no—
247#395—141no—
248#389—71no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—141no—
253#387—141no—
254#385—141no—
255#384—71no—
256#384—71no—
257#380—71no—
258#380—71no—
259C4×C22⋊C4C2×D711yescharacteristic
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#336C2×D711yescharacteristic
267#336C2×D711yescharacteristic
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7C4⋊Q812yes—
275C23×C7C22×D411yescharacteristic
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7C4⋊Q812yes—
293C2×C4×C7AGL(1,ℤ/8)12yes—
294C2×C4×C7#13712yes—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—84no—
326Dic14—84no—
327C7×C8—22no—
328C7×C8—22no—
329C7×C8—22no—
330C7×C8C22⋊Q814yescomplemented
331C22×Q8—141no—
332C23×C4—141no—
333C23×C4C22×D711yescharacteristic
334C2×M16—22no—
335C2×M16—22no—
336C2×M16Dic1414yescomplemented
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C4⋊Q8—142no—
342C4⋊Q8—281no—
343#125—142no—
344#125—281no—
345C22⋊Q8—284no—
346#117—142no—
347#117—281no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—281no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4D2811yescharacteristic
374C2×C22⋊C4D2811yescharacteristic
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D711yescharacteristic
378#107—74no—
379#107—142no—
380#107—142no—
381#107—142no—
382#106—142no—
383#106—142no—
384#106—142no—
385#106—142no—
386#106—281no—
387#106—281no—
388C22⋊C8Dic1414yescomplemented
389#97—22no—
390C4×C8—22no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95D2811yescharacteristic
398#95D2811yescharacteristic
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7C2×C4⋊Q811yescharacteristic
407C22×C7D4211yescharacteristic
408C22×C7#54711yescharacteristic
409C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
410C22×C7C2×#13711yescharacteristic
411C22×C7#57711yescharacteristic
412C22×C7#57711yescharacteristic
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7C2×C22⋊Q811yescharacteristic
419C4×C7#54411yescharacteristic
420C4×C7D4×Q811yescharacteristic
421C4×C7#55411yescharacteristic
422Dic7—42no—
423Dic7—42no—
424Dic7—81no—
425Dic7—81no—
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24C2×D2811yescharacteristic
431C2×Q8—282no—
432C2×Q8—282no—
433C2×Q8—561no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4—561no—
449C22×C4C2×Dic1412yes—
450C22×C4C2×Dic1412yes—
451C22×C4C2×D2811yescharacteristic
452C22×C4C2×D2811yescharacteristic
453C22×C4C2×D2811yescharacteristic
454C22×C4C2×D2811yescharacteristic
455C22×C4C2×D2811yescharacteristic
456C22×C4C23×D711yescharacteristic
457M16—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8C2×Dic1412yes—
465C2×C8C2×Dic1412yes—
466C2×C8C2×Dic1412yes—
467C2×C8C2×Dic1412yes—
468C2×C8D4×D712yes—
469C2×C8#115912yes—
470C4⋊C4—72no—
471C4⋊C4—72no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—282no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C4⋊C4—561no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—282no—
501C22⋊C4—282no—
502C22⋊C4—561no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—282no—
508C2×C7—42no—
509C2×C7#244011yescharacteristic
510C2×C7#316311yescharacteristic
511C2×C7#316511yescharacteristic
512C2×C7#327211yescharacteristic
513C2×C7#327811yescharacteristic
514C2×C7#334411yescharacteristic
515C2×C7#334511yescharacteristic
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23#643212yes—
523C23C22×D2811yescharacteristic
524Q8—564no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—564no—
566C2×C4#643212yes—
567C2×C4#647112yes—
568C2×C4#647212yes—
569C2×C4C22×Dic1411yescharacteristic
570C2×C4C22×Dic1411yescharacteristic
571C2×C4C2×D4×D711yescharacteristic
572C2×C4C2×#115911yescharacteristic
573C2×C4#655311yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—22no—
577C8—22no—
578C8#644114yescomplemented
579C7#2607811yescharacteristic, complemented, Sylow
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#643211yescharacteristic
588C22D4×D2811yescharacteristic
589C22#9024611yescharacteristic
590C22C2×#647111yescharacteristic
591C22C2×#647211yescharacteristic
592C22#9044211yescharacteristic
593C22#9044211yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—282no—
600C4—282no—
601C4—561no—
602C4—561no—
603C4—561no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4C2×#644111yescharacteristic
608C4#9021011yescharacteristic
609C4D4×Dic1411yescharacteristic
610C4#9023411yescharacteristic
611C2—42no—
612C2#1173093811yescharacteristic
613C2#1173595111yescharacteristic
614C2#1173599011yescharacteristic
615C2#1173691511yescharacteristic
616C2#1173691611yescharacteristic
617C2#1173699311yescharacteristic
618C2#1173700511yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1753C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1753D7
hypocenterC7#26078
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#89474C2semidirect
9. #11727573C2semidirect
10. #11727524C2semidirect
11. #11727518C2semidirect
12. #11726394C2semidirect
13. #11726393C2semidirect
14. #11726388C2semidirect
15. #11726387C2semidirect
67. C2×C7×M16Q8semidirect
109. C7×C22⋊C8Q8semidirect
330. C7×C8C22⋊Q8semidirect
336. C2×M16Dic14semidirect
388. C22⋊C8Dic14semidirect
578. C8#6441semidirect
579. C7#26078semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#898932
C2×#894742
#117309392
#117309382
C7×#17532
#117297422
#117276832
#117276822
#117275732
#117275242
#117275182
#117263942
#117263932
#117263882
#117263872
#260787